Enumeration Of Classes Research Articles

Model fit criteria curve behaviour in class enumeration – a diagnostic tool for model (mis)specification in longitudinal mixture modelling

The use of longitudinal finite mixture models (FMMs) to identify latent classes of individuals following similar paths of temporal development is gaining traction in applied research. However, FMM’s users may be unaware of how data features as well as the inappropriate specification of the model’s covariance structure impacts class enumeration. To elucidate this, we investigated model fit-criteria curve behaviour across an array of data conditions and covariance structures. Fit statistic patterns were variable among the fit criteria and across a range of data conditions. This variability was greatly attributable to the level of class separation and the presence/absence of random effects. Our findings support some widely held notions (e.g. BIC outperforms other criteria) while debunking others (adding random effects is not always the solution). Based on the obtained results, we present guidelines on how the behaviour of fit criteria curves can be used as a diagnostic aid during class enumeration.

Mixed Effects of Item Parceling on Performance of Factor Mixture Modeling

ABSTRACT Factor mixture modeling (FMM) is generally complex with both unobserved categorical and unobserved continuous variables. We explore the potential of item parceling to reduce the model complexity of FMM and improve convergence and class enumeration accordingly. To this end, we conduct Monte Carlo simulations with three types of data, continuous, polytomous, and binary under two levels of model complexity, constrained FMM under strict invariance and relaxed FMM under scalar or metric invariance. The results show that item parceling could be advantageous for FMM with binary items but not with continuous or polytomous items.

Branching rules and commuting probabilities for Triangular and Unitriangular matrices

This paper concerns the enumeration of simultaneous conjugacy classes of [Formula: see text]-tuples of commuting matrices in the upper triangular group [Formula: see text] and unitriangular group [Formula: see text] over the finite field [Formula: see text] of odd characteristic. This is done for [Formula: see text] and [Formula: see text], by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities [Formula: see text] for [Formula: see text] in each case.

Comparison of Three Approaches to Class Enumeration in Growth Mixture Modeling when Time Structures are Variant Across Latent Classes

ABSTRACT In conventional approaches to Growth Mixture Modeling (GMM), a trajectory is first estimated using latent growth curve modeling that serves as a baseline trajectory for the GMM. In this approach, time structures are held invariant across latent classes when identifying the number of latent classes. However, this popular way of conducting GMM could undermine a proper estimation, especially under the condition where a distinct trajectory exists for different classes in the population. This study compared the class enumeration performance in a conventional GMM against two alternatives in which latent classes do not take the same functional forms of change across time: (1) Unstructured Mixture Models (UMM) and (2) Latent Basis Models (LBM). Results revealed that the UMM performs well when one latent class takes a different shape of growth. Based on various design conditions, the relative performance of the three approaches in terms of class enumeration is examined and discussed.

Examining the Fairness of the University Entrance Exam: A Latent Class Analysis Approach to Differential Item Functioning

Measurement has been ubiquitous in all areas of education for at least a century. Various methods have been suggested to examine the fairness of education tests especially in high-stakes contexts. The present study has adopted the newly proposed ecological approach to differential item functioning (DIF) to investigate the fairness of the Iranian nationwide university entrance exam. To this end, the actual data from an administration of the test were obtained and analyzed through both traditional logistic regression and latent class analysis (LCA) techniques. The initial DIF analysis through logistic regression revealed that 19 items (out of 70) showed either uniform or non-uniform DIF. Further examination of the sample through LCA showed that the sample is not homogeneous. LCA class enumeration revealed that three classes can be identified in the sample. DIF analysis for separate latent classes showed that three serious differences in the number of DIF items identified in each latent class ranging from zero items in latent class 3 to 43 items in latent class 2. The inclusion of the covariates in the model also showed that latent class membership could be significantly predicted from high school GPA, field of study, and acceptance quota. It is argued that the fairness of the test might be under question. The implications of the findings for the validity of the test are discussed in detail.

Investigating the Impact of Covariate Inclusion on Sample Size Requirements of Factor Mixture Modeling: A Monte Carlo Simulation Study

ABSTRACT Factor mixture modeling (FMM) has been increasingly used in behavioral and social sciences to capture underlying population heterogeneity. This Monte Carlo simulation study investigated a common issue in applications of FMM, sample size requirements, particularly taking into account the impact of covariate inclusion. Results indicated that it was critical to include a proper covariate into FMM for accurate class enumeration and the required sample size became smaller with larger covariate effect. Overall, the statistical power for detecting factor mean difference was adequate but positive bias was observed in some conditions. The required minimum sample sizes were suggested for applied researchers.

Enumeration of Permutation Classes and Weighted Labelled Independent Sets

In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.

Robustness of Latent Profile Analysis to Measurement Noninvariance Between Profiles.

Latent profile analysis (LPA) identifies heterogeneous subgroups based on continuous indicators that represent different dimensions. It is a common practice to measure each dimension using items, create composite or factor scores for each dimension, and use these scores as indicators of profiles in LPA. In this case, measurement models for dimensions are not included and potential noninvariance across latent profiles is not modeled in LPA. This simulation study examined the robustness of LPA in terms of class enumeration and parameter recovery when the noninvariance was unmodeled by using composite or factor scores as profile indicators. Results showed that correct class enumeration rates of LPA were relatively high with small degree of noninvariance, large class separation, large sample size, and equal proportions. Severe bias in profile indicator mean difference was observed with intercept and loading noninvariance, respectively. Implications for applied researchers are discussed.

Symmetric peaks and symmetric valleys in Dyck paths

The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Flórez and Rodríguez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving multivariate generating functions that keep track of the number of symmetric peaks and the number of asymmetric peaks, as well as the widths of these peaks. We recover a formula of Denise and Simion as a special case of our results.We also consider the analogous but more intricate notion of symmetric valleys. We find a continued fraction expression for the generating function of Dyck paths with respect to the number of symmetric valleys and the sum of their widths, which provides an unexpected connection between symmetric valleys and statistics on ordered rooted trees. Finally, we enumerate Dyck paths whose peak or valley heights satisfy certain monotonicity and unimodality conditions, using a common framework to recover some known results, and relating our questions to the enumeration of certain classes of column-convex polyominoes.

Exploring Class Enumeration in Bayesian Growth Mixture Modeling Based on Conditional Medians

Growth mixture modeling is a popular analytic tool for longitudinal data analysis. It detects latent groups based on the shapes of growth trajectories. Traditional growth mixture modeling assumes that outcome variables are normally distributed within each class. When data violate this normality assumption, however, it is well documented that the traditional growth mixture modeling mislead researchers in determining the number of latent classes as well as in estimating parameters. To address nonnormal data in growth mixture modeling, robust methods based on various nonnormal distributions have been developed. As a new robust approach, growth mixture modeling based on conditional medians has been proposed. In this article, we present the results of two simulation studies that evaluate the performance of the median-based growth mixture modeling in identifying the correct number of latent classes when data follow the normality assumption or have outliers. We also compared the performance of the median-based growth mixture modeling to the performance of traditional growth mixture modeling as well as robust growth mixture modeling based on t distributions. For identifying the number of latent classes in growth mixture modeling, the following three Bayesian model comparison criteria were considered: deviance information criterion, Watanabe-Akaike information criterion, and leave-one-out cross validation. For the median-based growth mixture modeling and t-based growth mixture modeling, our results showed that they maintained quite high model selection accuracy across all conditions in this study (ranged from 87 to 100%). In the traditional growth mixture modeling, however, the model selection accuracy was greatly influenced by the proportion of outliers. When sample size was 500 and the proportion of outliers was 0.05, the correct model was preferred in about 90% of the replications, but the percentage dropped to about 40% as the proportion of outliers increased to 0.15.

Improving convergence in growth mixture models without covariance structure constraints.

Growth mixture models are a popular method to uncover heterogeneity in growth trajectories. Harnessing the power of growth mixture models in applications is difficult given the prevalence of nonconvergence when fitting growth mixture models to empirical data. Growth mixture models are rooted in the random effect tradition, and nonconvergence often leads researchers to modify their intended model with constraints in the random effect covariance structure to facilitate estimation. While practical, doing so has been shown to adversely affect parameter estimates, class assignment, and class enumeration. Instead, we advocate specifying the models with a marginal approach to prevent the widespread practice of sacrificing class-specific covariance structures to appease nonconvergence. A simulation is provided to show the importance of modeling class-specific covariance structures and builds off existing literature showing that applying constraints to the covariance leads to poor performance. These results suggest that retaining class-specific covariance structures should be a top priority and that marginal models like covariance pattern growth mixture models that model the covariance structure without random effects are well-suited for such a purpose, particularly with modest sample sizes and attrition commonly found in applications. An application to PTSD data with such characteristics is provided to demonstrate (a) convergence difficulties with random effect models, (b) how covariance structure constraints improve convergence but to the detriment of performance, and (c) how covariance pattern growth mixture models may provide a path forward that improves convergence without forfeiting class-specific covariance structures.

Latent Class Trajectory and Growth Mixture Models in the Study of Physical Resilience

After a stressor, individuals may experience different trajectories of function and recovery. One potential explanation for this variation is differing trajectories may be indicators of differing classes or levels of resilience to the stressor. Latent Class Trajectory (LCTA) and Growth Mixture models (GMM) are two similar approaches used to discover the number and types of trajectories in a study population. Class membership may determine the shape and level of recovery, which may be predicted by individual characteristics. In this talk, we present some insights to using these models to successfully identify the number of classes of trajectories, membership of trajectory classes, and the functional form of the trajectory. We will identify methods for deciding class enumeration, indices for assessing fit quality, and, importantly, the importance of proper model specification. Real life and simulated examples will be shown to compare and contrast differences between GMM and LCTA results. Part of a symposium sponsored by Epidemiology of Aging Interest Group.

Exploring the Enumeration Accuracy of Cross-Validation Indices in Latent Class Analysis

ABSTRACT A crucial issue when estimating mixture models is selecting the model with the correct number of latent classes underlying the data, which is commonly referred to as class enumeration. Although cross-validation methods have been suggested with mixture models to help augment the class enumeration process (Masyn, 2013), they have been seldom used. The purpose of this simulation study was to compare the performance of traditionally used single sample enumeration indices with the performance of cross-validation indices when selecting the correct latent class model with binary indicators. Various conditions were manipulated, including the number of indicators, sample size, class separation, mixing proportions, and number of latent classes. The enumeration accuracy of sixteen indices (traditional, cross-validated, and double cross-validated) were documented in the manipulated conditions. The traditional sample-size adjusted BIC index was the most accurate among the indices. The performance of the double cross-validated -2LL was also promising. Recommendations are provided.

Refined enumeration of symmetry classes of alternating sign matrices

We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.

Doubly Λ-commuting row isometries, universal models, and classification

The goal of the paper is to study the structure of the k-tuples of doubly Λ-commuting row isometries and the C⁎-algebras they generate from the point of view of noncommutative multivariable operator theory. We obtain Wold decompositions, in this setting, and use them to classify the k-tuples of doubly Λ-commuting row isometries up to a unitary equivalence. We prove that there is a one-to-one correspondence between the unitary equivalence classes of k-tuples of doubly Λ-commuting row isometries and the enumerations of 2k unitary equivalence classes of unital representations of the twisted Λ-tensor algebras ⊗i∈AcΛOni, as A is any subset of {1,…,k}, where Oni is the Cuntz algebra with ni generators. In addition, we obtain a description and parametrization of the irreducible k-tuples of doubly Λ-commuting row isometries.We introduce the standard k-tuple S:=(S1,…,Sk) of doubly Λ-commuting pure row isometries Si:=[Si,1⋯Si,ni] acting on the Hilbert space ℓ2(Fn1+×⋯×Fnk+), where Fn+ is the unital free semigroup with n generators, and prove that the universal C⁎-algebra generated by a k-tuple of doubly Λ-commuting row isometries is ⁎-isomorphic to the C⁎-algebra C⁎({Si,s}). We introduce the regular Λ-polyball BΛ(H) and show that a k-tuple T:=(T1,…,Tk) of row operators Ti:=[Ti,1…Ti,ni], acting on H, admits S as universal model, i.e. there is a Hilbert space D such that H is jointly co-invariant for Si,s⊗ID andTi,s⁎=(Si,s⁎⊗ID)|H,i∈{1,…,k} and s∈{1,…,ni}, if and only if T is a pure element of BΛ(H). This leads to von type Neumann inequalities and the introduction of the noncommutative Berezin transform associated with the elements of the regular Λ-polyball, which plays an important role. We use the Berezin kernel to obtain a characterization of the Beurling type jointly invariant subspaces of the standard k-tuple S=(S1,…,Sk) and provide a classification result for the pure elements in the regular Λ-polyball.We show that any k-tuple T in the regular Λ-polyball admits a minimal dilation which is a k-tuple of doubly Λ-commuting row isometries and is uniquely determined up to an isomorphism. Using the Wold decompositions obtained, we show that T is a pure element in BΛ(H) if and only if its minimal dilation is a pure element in BΛ(K). In the particular case when n1=⋯=nk=1, we obtain an extension of Brehmer's result, showing that any k-tuple in the regular Λ-polyball admits a unique minimal doubly Λ-commuting unitary dilation.

Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C80, C180, and C240

We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C80 to C240. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Möbius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C80 with a chamfered dodecahedron structure, icosahedral C180, and C240 with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

Testing Measurement Invariance Across Unobserved Groups: The Role of Covariates in Factor Mixture Modeling.

Factor mixture modeling (FMM) has been increasingly used to investigate unobserved population heterogeneity. This study examined the issue of covariate effects with FMM in the context of measurement invariance testing. Specifically, the impact of excluding and misspecifying covariate effects on measurement invariance testing and class enumeration was investigated via Monte Carlo simulations. Data were generated based on FMM models with (1) a zero covariate effect, (2) a covariate effect on the latent class variable, and (3) covariate effects on both the latent class variable and the factor. For each population model, different analysis models that excluded or misspecified covariate effects were fitted. Results highlighted the importance of including proper covariates in measurement invariance testing and evidenced the utility of a model comparison approach in searching for the correct specification of covariate effects and the level of measurement invariance. This approach was demonstrated using an empirical data set. Implications for methodological and applied research are discussed.

Class enumeration false positive in skew-t family of continuous growth mixture models.

Growth Mixture Modeling (GMM) has gained great popularity in the last decades as a methodology for longitudinal data analysis. The usual assumption of normally distributed repeated measures has been shown as problematic in real-life data applications. Namely, performing normal GMM on data that is even slightly skewed can lead to an over selection of the number of latent classes. In order to ameliorate this unwanted result, GMM based on the skew t family of continuous distributions has been proposed. This family of distributions includes the normal, skew normal, t, and skew t. This simulation study aims to determine the efficiency of selecting the “true” number of latent groups in GMM based on the skew t family of continuous distributions, using fit indices and likelihood ratio tests. Results show that the skew t GMM was the only model considered that showed fit indices and LRT false positive rates under the 0.05 cutoff value across sample sizes and for normal, and skewed and kurtic data. Simulation results are corroborated by a real educational data application example. These findings favor the development of practical guides of the benefits and risks of using the GMM based on this family of distributions.

Enumeration and Generation of $\rm PSL$ Equivalence Classes for Quad-Phase Codes of Odd Length

Quad-phase codes of a given length can be grouped into equivalence classes based on operations preserving autocorrelation peak sidelobe level, or “ ${\rm PSL}$ -preserving operators.” The task of enumerating these equivalence classes is facilitated by establishing a relationship with the problem of enumerating equivalence classes of $2\times N$ binary grids with respect to a pair of binary grid symmetries. We show this connection by a one-to-one mapping between even-length quad-phase codes and $2\times N$ binary grids. When $N$ is even, this mapping allows the known enumeration of $2\times N$ binary grids to be applied to the enumeration of even-length quad-phase code PSL equivalence classes. Furthermore, this equivalence instructs the development of a simpler algorithm for visiting single representatives of quad-phase-code equivalence classes relative to the operations that preserve PSL. Lastly, we conduct exhaustive searches to determine the optimal ${\rm PSL}$ for quad-phase codes of lengths 25 and 26 (for the smaller of these lengths, the search space contains $4^{25}=2^{50}$ , or roughly 1 quadrillion codes). One of the devices used to narrow the search space while achieving an exhaustive search is to seed the search with the equivalence class representatives of length $N=6$ . The optimal ${\rm PSL}$ for length-25 quad-phase codes is determined to be 2, achieved by the codes in a single size-32 equivalence class. For length 26, the optimal ${\rm PSL}$ is found to be $\sqrt{5}$ , achieved by $604\,480$ codes belonging to 9445 equivalence classes of size 64.

Reconsidering Multilevel Latent Class Models: Can Level-2 Latent Classes Affect Item Response Probabilities?

Multilevel latent class analysis (MLCA) has been increasingly used to investigate unobserved population heterogeneity while taking into account data dependency. Nonparametric MLCA has gained much popularity due to the advantage of classifying both individuals and clusters into latent classes. This study demonstrated the need to relax the assumption in specifying the nonparametric MLCA: item response probabilities varied only across level-1 latent classes, but not level-2 latent classes. An empirical demonstration with data from the Trends in International Mathematics and Science Study (TIMSS) 2011 showed that item response probabilities could vary across both level-1 and level-2 latent classes. This relaxed MLCA yielded better model fit and provided more nuanced understanding of the heterogeneous response patterns. Monte Carlo simulation was conducted to evaluate class enumeration and assignment accuracy of the relaxed MLCA. Based on the simulation results, we recommended the use of AIC in class enumeration and highlighted the benefits of having larger cluster size.